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Strong consistency and rates for recursive probability density estimators of stationary processes

Author

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  • Masry, Elias
  • Györfi, László

Abstract

Let {Xj}j = - [infinity][infinity] be a vector-valued stationary process with a first-order univariate probability density f on Rd. We consider the recursive estimation of f(x) from n observations {Xj}j=1n which need not be independent. For processes {Xj}j = - [infinity][infinity] which are asymptotically uncorrelated, we establish sharp rates for the almost sure convergence of kernel-type estimators fn(x).

Suggested Citation

  • Masry, Elias & Györfi, László, 1987. "Strong consistency and rates for recursive probability density estimators of stationary processes," Journal of Multivariate Analysis, Elsevier, vol. 22(1), pages 79-93, June.
    • Handle: RePEc:eee:jmvana:v:22:y:1987:i:1:p:79-93
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    Citations

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    Cited by:

    1. Marc Hallin & Lanh Tran, 1996. "Kernel density estimation for linear processes: Asymptotic normality and optimal bandwidth derivation," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 48(3), pages 429-449, September.
    2. Carbon, Michel & Tran, Lanh Tat & Wu, Berlin, 1997. "Kernel density estimation for random fields (density estimation for random fields)," Statistics & Probability Letters, Elsevier, vol. 36(2), pages 115-125, December.
    3. Lacour, Claire, 2008. "Nonparametric estimation of the stationary density and the transition density of a Markov chain," Stochastic Processes and their Applications, Elsevier, vol. 118(2), pages 232-260, February.
    4. Sophie Dabo-Niang & Anne-Françoise Yao, 2013. "Kernel spatial density estimation in infinite dimension space," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 76(1), pages 19-52, January.
    5. Michel Carbon, 2008. "Asymptotic Normality of Frequency Polygons for Random Fields," Working Papers 2008-09, Center for Research in Economics and Statistics.
    6. Lu, Zudi & Chen, Xing, 2004. "Spatial kernel regression estimation: weak consistency," Statistics & Probability Letters, Elsevier, vol. 68(2), pages 125-136, June.
    7. Carbon, Michel & Garel, Bernard & Tran, Lanh Tat, 1997. "Frequency polygons for weakly dependent processes," Statistics & Probability Letters, Elsevier, vol. 33(1), pages 1-13, April.
    8. Hallin, Marc & Lu, Zudi & Tran, Lanh T., 2004. "Kernel density estimation for spatial processes: the L1 theory," Journal of Multivariate Analysis, Elsevier, vol. 88(1), pages 61-75, January.
    9. Michel Carbon, 2005. "Frequency Polygons for Random Fields," Working Papers 2005-04, Center for Research in Economics and Statistics.
    10. Senoussi, R., 2000. "Uniform iterated logarithm laws for martingales and their application to functional estimation in controlled Markov chains," Stochastic Processes and their Applications, Elsevier, vol. 89(2), pages 193-211, October.
    11. P. Cattiaux & José R. León & C. Prieur, 2015. "Recursive estimation for stochastic damping hamiltonian systems," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 27(3), pages 401-424, September.

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